Solve for $x$ and $y$ using elimination. $\begin{align*}-7x+6y &= 1 \\ -x+4y &= -1\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $3$ $\begin{align*}14x-12y &= -2\\ -3x+12y &= -3\end{align*}$ Add the top and bottom equations. $11x = -5$ Divide both sides by $11$ and reduce as necessary. $x = -\dfrac{5}{11}$ Substitute $-\dfrac{5}{11}$ for $x$ in the top equation. $-7( -\dfrac{5}{11})+6y = 1$ $\dfrac{35}{11}+6y = 1$ $6y = -\dfrac{24}{11}$ $y = -\dfrac{4}{11}$ The solution is $\enspace x = -\dfrac{5}{11}, \enspace y = -\dfrac{4}{11}$.